Population history shapes human biology, genetic variation, and disease risk. Despite its importance, analytical methods to study history on a genome-wide scale are still limited in both their resolution and qualitative ability to reconstruct aspects of the past. This makes it a prioriy to develop methods that can analyze history in the large data sets that are now practical to generate. These new methods also need to deal with the major population mixture events that have been documented in the last five years as a result of the genomic and ancient DNA data revolutions. Thus, it is no longer adequate to simply propose models of population relationships without mixture. Instead, the hypotheses that need to be tested for consistency with data are ones that involve major, already-known mixture events. The toolkit for studying the historical events-whilst taking into account known admixture events-is at present inadequate. This proposal aims to build on and extend the work of R01 GM100233, which from 2012-2015 supported the joint PIs' central research program on developing methods for studying human population history, resulting in 31 publications directly linked to the grant. We now propose four new Aims: (1) To develop tests for population mixture that work even without explicit phylogenetic models; (2) To build a comprehensive toolkit for studying population history using linkage disequilibrium; (3) To capitalize on the power of the joint allele frequency spectrum for studies of human history; (4) To enable powerful comparisons of X and the autosomes to reveal sex-biased demography. This grant will be of value in three ways. (a) It will support the development of methods and user- friendly software that will be important for both evolutionary and medical genetics. (b) It will support work that will result in insights relevant to finding disease genes in human populations that are recently or anciently admixed. (c) It will lead to new discoveries about human history. The link to medical genetics is important. In the past, we have been successful at drawing a direct connection between our laboratory's work on detecting and characterizing population mixture, and human biology including genetic susceptibility to disease. We leveraged the history of admixture in African Americans to make new disease gene discoveries (for example, risk factors for prostate cancer), to understand variation in disease ris across populations, and to document differences in the biology of recombination between African Americans and people who do not have West African ancestry, which is predicted to lead to different levels of risk for diseases associated with errors in recombination. We anticipate that the approaches for understanding and modeling population history that we develop with the support of this grant will continue to synergize with the latest research on human genetic variation and disease risk.